``We bent a fair amount, but we didn't break,'' Anaheim head coach Dallas Eakins said.
Defenceman Jamie Drysdale scored the OT winner at 3:26 tapping in a Troy Terry pass.
``Troy kind of did all the work, made a great fake, great curl and drag and then slipped it through the guy, right on my tape back door,'' Drysdale said.
It was the first OT winner for the 19-year-old from Toronto.
``Awesome,'' Drysdale declared. ``It's obviously really exciting, especially being back here in Canada. It's nice, even though the fans were cheering against us, the energy was real good. It's nice to have fans back in the building.''
Cam Fowler and Rickard Rakell also scored for the Ducks (2-1-0), who won the first of four straight road games.
Anaheim goaltender John Gibson made 41 saves for the win after sitting out Friday's loss to Minnesota with a lower-body injury.
``To come out with this win, not only just being my first goal in overtime, this was a really big win for our group and it'll set us up nicely moving forward into this road trip,'' Drysdale said.
Blake Coleman scored in his Flames debut and Elias Lindholm also had a goal for Calgary (0-1-1).
Johnny Gaudreau had two assists, while Flames goaltender Jacob Markstrom stopped 24 shots in the loss.
Calgary lost a second straight game putting 40-plus shots on the opposition's net. The Flames generated 47 in a 5-2 season-opening loss in Edmonton on Saturday.
``We have 40 shots in both games, but we've got to get someone in front of the net,'' Gaudreau said. ``We've got to take the goalie's eyes away and find a way to score some greasy goals.''
With 15,174 in attendance, Scotiabank Saddledome wasn't full for the Flames' first regular-season game with spectators since March 8, 2020.
The COVID-19 pandemic had the Flames playing in front of empty seats in the 19,289-seat arena until the pre-season.
Ducks defenceman Josh Manson took a high-sticking penalty with just under four minutes remaining in regulation, but Calgary's power play couldn't produce a go-ahead goal.
Rakell on a give and go with Adam Henrique pulled Anaheim even at 12:55 of the third period.
Henrique fed an unchecked Rakell in the high slot for the latter to whip the puck over Markstrom's right pad.
Calgary led 2-1 at 3:52 of the second period on Lindholm's second in as many games.
He collected a cross-ice pass from Gaudreau and threaded a shot between Gibson's pads from the faceoff dot.
Calgary's Milan Lucic and Anaheim's Nicolas Deslauriers fought following a faceoff midway through the second period.
Fowler pulled the visitors even with a power-play goal at 15:54 of the opening period. He wristed a drop pass from Kevin Shattenkirk by Markstrom's right pad.
Anaheim had the man advantage because Matthew Tkachuk chopped an airborne puck down while standing on Calgary's bench, and was called for interference.
Coleman scored the season's first goal at the Saddledome at 7:43. Gaudreau tapped the puck up to the winger in the neutral zone. Coleman generated a shooting lane with his speed and fired a wrist shot under Gibson's left arm.
Winner of back-to-back Stanley Cups with the Tampa Bay Lightning, Coleman signed a six-year, US$29.4-million contract with Calgary in the off-season.
The Flames start a five-game road trip Thursday in Detroit.
Notes: Calgary is without a captain. The Flames are rotating assistant captaincies among four players: Tkachuk, Mikael Backlund, Sean Monahan and Chris Tanev. ? Former Flames head coach Geoff Ward, now a Ducks assistant coach, returned to the Saddledome for the first time since Calgary fired him March 4. ? Mason McTavish, who at age 18 is Anaheim's youngest-ever goal scorer, didn't play Monday's third period because of a lower-body injury.